Non-conservative system and velocity dependent potentials

Question

I'm studying Lagrangian mechanics, but I'm a little bit upset because when dealing with Lagrange's equations, we mostly consider conservative systems. If the system is non conservative they are very brief by saying that 'sometimes' there exist a velocity dependent potential $U(q,\dot{q},t)$ such that the generalized force $Q_j$ of the standard system can be written in terms of this potential. $$Q_j =\frac{d}{dt}\left(\frac{\partial U}{\partial \dot{q}_j}\right)-\frac{\partial U}{\partial q_j}$$ They give as example, charged particles in a static EM field.

1. But my question is, if we can find this velocity dependent potential for any generalized force?

2. If not, we can't use Lagrangian mechanics?

Answer 1

1. No, (generalized) velocity dependent potentials $U(q,\dot{q},t)$ do not exist for all (generalized) forces $Q_j$. See e.g. this Phys.SE post.

2. Even if no variational formulation exists, one may still consider Lagrange equations, cf. e.g. this Phys.SE post.